There has been recent interest in phrasing such expression languages
generically, obtaining their syntax and semantics as well as
meta-theoretic results including Kleene theorems by instantiation of a
parametrized framework. This is achieved by abstracting the type of
systems as coalgebras for a given type functor. This line of work
originates with expression languages for a specific class of functors
that essentially covers relational systems, so-called Kripke
polynomial functors [1]}, and was subsequently extended
to cover also weighted systems [2]}. A generic expression
language for arbitrary finitary functors can be based on algebraic
functor presentations [3]}. Here, we introduce a similar and, as
it will turn out, in fact largely equivalent generic expression
language for finitary functors, which we base on coalgebraic
modalities in predicate lifting style, following the paradigm of
coalgebraic logic [4]}; on predicate liftings, we impose
strong conditions, notably including preservation of singletons. Marti
and Venema [5]} have shown that for functors admitting a
lax extension (in particular for functors that admit a
separating set of monotone predicate liftings), one can convert
operations from the functor presentation into predicate liftings, the
so-called Moss liftings. We show that the Moss liftings
preserve singletons; the converse does not hold in general, i.e. not
all singleton-preserving predicate liftings are Moss liftings under a
given lax extension.
